On a semi-stochastic model arising in a biological system

Abstract

A stochastic version of a mathematical model for chemotherapy, which was developed deterministically by Bellman et al. [5, 6], is given. After an injection at the heart entrance, the concentration of drug in the blood plasma of a biological system with one organ and a simplified heart results in the semistochastic integral equation u L(t;ω) = − c V ∗ ∫ 0 t [u L(y;ω) − u R(y;ω)] dy, t ⩽ 0, where u L ( t; ω) is the concentration of drug plasma leaving the heart and u R ( t; ω) is the concentration of drug in plasma entering the heart at time t. The function u L ( t; ω) is a deterministic function of time for 0 ⩽ t < T, where T is the blood recirculation time lag, and a random function of t, T ⩽ t ⩽ M, for some M ∈ ( T ∞). It is shown that the integral equation has a solution for both t ∈ [0, T), using the method of successive approximations, and t ∈ [ T, M], using some methods of probabilistic functional analysis.